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#### All Chapter 3 Marks

12th Standard EM

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Maths

Time : 02:00:00 Hrs
Total Marks : 144
48 x 3 = 144
1. Find the inverse of the non-singular matrix A =  $\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right]$, by Gauss-Jordan method.

2. Solve the following system of linear equations by matrix inversion method:
2x + 5y = −2, x + 2y = −3

3. Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

4. Verify that (A-1)T = (AT)-1 for A=$\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right]$.

5. If z1=3,z2=-7i, and z3=5+4i, show that z1(z2+z3)=z1z2+z1z3

6. The complex numbers u,v, and w are related by $\cfrac { 1 }{ u } =\cfrac { 1 }{ v } +\cfrac { 1 }{ w }$ If v=3−4i and w=4+3i, find u in rectangular form.

7. Explain the falacy:

8. Find the locus of z if Re$\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right)$ =0.

9. If α and β are the roots of the quadratic equation 2x2−7x+13 = 0 , construct a quadratic equation whose roots are α2 and β2.

10. Solve the equation x3-3x2-33x+35 = 0.

11. Find the number .of real solu,tlons of sin (ex) -5x + 5-x

12. Solve:(x-1)4+(x-5)4=82

13. Find the domain of the following
${ sin }^{ -1 }\left( \frac { { x }^{ 2 }+1 }{ 2x } \right)$

14. Solve
${ cot }^{ -1 }x-{ xot }^{ -1 }\left( x+2 \right) =\frac { \pi }{ 12 } ,x>0$

15. Evaluate $cos\left[ { sin }^{ -1 }\cfrac { 3 }{ 5 } +{ sin }^{ -1 }\cfrac { 5 }{ 13 } \right]$

16. Solve ${ tan }^{ -1 }\left( \cfrac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \cfrac { 1-{ x }^{ 2 } }{ 2x } \right) =\cfrac { \pi }{ 3 } ,x>0$

17. Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
$\frac { { \left( x-3 \right) }^{ 2 } }{ 225 } +\frac { { \left( y-4 \right) }^{ 2 } }{ 289 } =1$

18. The equation of the ellipse is $\frac { { \left( x-11 \right) }^{ 2 } }{ 484 } +\frac { { y }^{ 2 } }{ 64 } =1$.  ( x and y are measured in centimeters) where to the nearest centimeter, should the patient’s kidney stone be placed so that the reflected sound hits the kidney stone?

19. Find the circumference and area of the circle x2 +y2 - 2x + 5y + 7 = 0

20. For the hyperbola 3x2 - 6y2 = -18, find the length of transverse and conjugate axes and eccentricity.

21. Find the altitude of a parallelepiped determined by the vectors $\vec { a } =2\hat { i } +5\hat { j } +3\hat { k }$$\hat { b } =\hat { i } +3\hat { j } -2\hat { k }$ and $\vec { c } =-3\vec { i } +\vec { j } +4\vec { k }$ if the base is taken as the parallelogram determined by $\vec { b }$ and $\vec { c }$

22. Find the equation of the plane passing through the intersection of the planes 2x+3y−z+7=0 and andx+y−2z+5=0 and is perpendicular to the planex+y−3z−5=0.

23. Find the Cartesian form of the equation of the plane $\overset { \rightarrow }{ r } =\left( s-2t \right) \overset { \wedge }{ i } +\left( 3-t \right) \overset { \wedge }{ j } +\left( 2s+t \right) \overset { \wedge }{ k }$

()

s, t

24. Show that the four points whose position vectors are $6\overset { \wedge }{ i } -7\overset { \wedge }{ j } ,16\overset { \wedge }{ i } -29\overset { \wedge }{ j } -4\overset { \wedge }{ k } ,3\overset { \wedge }{ i } -6\overset { \wedge }{ j }$ are co-planar

()

1

25. Find the equations of tangent and normal to the curve y = x2 + 3x − 2 at the point (1, 2)

26. Using mean value theorem prove that for, a > 0, b > 0, le-a - e-bl < la - bl.

27. The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.

28. Evaluate the following limits, if necessary use L’Hopitals rule
(i) $\underset { x\rightarrow { 0 }^{ + } }{ lim } { x }^{ sinx }$
(ii) $\underset { x\rightarrow 0 }{ lim } \cfrac { cotx }{ cot2x }$
(iii) $\underset { x\rightarrow \frac { { \pi }^{ - } }{ 2 } }{ lim } \left( tanx \right) ^{ cosx }$

29. If w(x, y) = x3 − xy + y2, x, y ∊ R, find the linear approximation for w at (1,−1)

30. Let g( x,y)= x3 - yx + sin(x+y), x(t) = e3t, y(t) = t2, t ∈ R. Find $\frac { dg }{ dt }$

31. If f = $\frac { x }{ { x }^{ 2 }+{ y }^{ 2 } }$ then show that = $x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y }$ = -f

32. Find the approximate value of $\left( \cfrac { 17 }{ 81 } \right) ^{ \frac { 1 }{ 4 } }$ using linear approximation.

33. Evaluate $\\ \int _{ 0 }^{ 1 }{ { e }^{ -2x }(1+x-{ 2x }^{ 3 })dx }$

34. Evaluate $\int _{ 0 }^{ 2a }{ { x }^{ 2 }\sqrt { 2ax-{ x }^{ 2 } } } dx$

35. If $f(x)=\left| \begin{matrix} x+1 & 2x+1 & 3x+1 \\ 2x+1 & 3x+1 & x+1 \\ 3x+1 & x+1 & 2x+1 \end{matrix} \right|$ ,then find $\int _{ 0 }^{ 1 }{ f(x)dx }$
[Hint:R2➝R2➝R1;R2➝R3➝R1]

36. Evaluate $\int _{ 0 }^{ 1 }{ x(1-x) } ^{ n }dx$

37. Show that y = ax + $\frac { b }{ x }$, x ≠ 0 is a solution of the differential equation x2 y" + xy' - y = 0.

38. Solve ${ y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } =xy\frac { dy }{ dx }$

39. Verify that y=-x-1 is a solution of the D.E (y-x)dy-(y2-x2)dx=0

40. Solve :(1+e2x)dy+(1+y2)exdx=0

41. Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images.

42. For the random variable X with the given probability mass function as below, find the mean and variance.
$f(x)=\begin{cases} \begin{matrix} \cfrac { 1 }{ 2 } e^{ -\frac { x }{ 2 } } & for\quad x>0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}$

43. Give any three properties on expectation and variance.

44. The distribution of a continuous random variable X in range (−3,3) is given by p.d.f

Verify that the area under the curve is unity.

45. Establish the equivalence property p ➝ q ≡ ㄱp ν q

46. Verify whether the following compound propositions are tautologies or contradictions or contingency
(p ∧ q) ¬ (p ∨ q)

47. Let G = {1, i,-1, -i} under the binary operation multiplication. Find the inverse of all the elements.

48. On the set Q of rational numbers, an operation * is defined as a*b=k(a+b) where k is a given non zero number. Is it associative